# Properties

 Label 68.1.f.a Level $68$ Weight $1$ Character orbit 68.f Analytic conductor $0.034$ Analytic rank $0$ Dimension $2$ Projective image $D_{4}$ CM discriminant -4 Inner twists $4$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,1,Mod(47,68)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 1]))

N = Newforms(chi, 1, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("68.47");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 68.f (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$0.0339364208590$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(i)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{4}$$ Projective field: Galois closure of 4.2.19652.1 Artin image: $C_4\wr C_2$ Artin field: Galois closure of 8.0.1257728.1

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + ( - i - 1) q^{5} - i q^{8} + i q^{9} +O(q^{10})$$ q + z * q^2 - q^4 + (-z - 1) * q^5 - z * q^8 + z * q^9 $$q + i q^{2} - q^{4} + ( - i - 1) q^{5} - i q^{8} + i q^{9} + ( - i + 1) q^{10} + q^{16} - q^{17} - q^{18} + (i + 1) q^{20} + i q^{25} + (i + 1) q^{29} + i q^{32} - i q^{34} - i q^{36} + ( - i - 1) q^{37} + (i - 1) q^{40} + ( - i + 1) q^{41} + ( - i + 1) q^{45} - i q^{49} - q^{50} + (i - 1) q^{58} + (i - 1) q^{61} - q^{64} + q^{68} + q^{72} + (i + 1) q^{73} + ( - i + 1) q^{74} + ( - i - 1) q^{80} - q^{81} + (i + 1) q^{82} + (i + 1) q^{85} + (i + 1) q^{90} + ( - i - 1) q^{97} + q^{98} +O(q^{100})$$ q + z * q^2 - q^4 + (-z - 1) * q^5 - z * q^8 + z * q^9 + (-z + 1) * q^10 + q^16 - q^17 - q^18 + (z + 1) * q^20 + z * q^25 + (z + 1) * q^29 + z * q^32 - z * q^34 - z * q^36 + (-z - 1) * q^37 + (z - 1) * q^40 + (-z + 1) * q^41 + (-z + 1) * q^45 - z * q^49 - q^50 + (z - 1) * q^58 + (z - 1) * q^61 - q^64 + q^68 + q^72 + (z + 1) * q^73 + (-z + 1) * q^74 + (-z - 1) * q^80 - q^81 + (z + 1) * q^82 + (z + 1) * q^85 + (z + 1) * q^90 + (-z - 1) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} - 2 q^{5}+O(q^{10})$$ 2 * q - 2 * q^4 - 2 * q^5 $$2 q - 2 q^{4} - 2 q^{5} + 2 q^{10} + 2 q^{16} - 2 q^{17} - 2 q^{18} + 2 q^{20} + 2 q^{29} - 2 q^{37} - 2 q^{40} + 2 q^{41} + 2 q^{45} - 2 q^{50} - 2 q^{58} - 2 q^{61} - 2 q^{64} + 2 q^{68} + 2 q^{72} + 2 q^{73} + 2 q^{74} - 2 q^{80} - 2 q^{81} + 2 q^{82} + 2 q^{85} + 2 q^{90} - 2 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - 2 * q^4 - 2 * q^5 + 2 * q^10 + 2 * q^16 - 2 * q^17 - 2 * q^18 + 2 * q^20 + 2 * q^29 - 2 * q^37 - 2 * q^40 + 2 * q^41 + 2 * q^45 - 2 * q^50 - 2 * q^58 - 2 * q^61 - 2 * q^64 + 2 * q^68 + 2 * q^72 + 2 * q^73 + 2 * q^74 - 2 * q^80 - 2 * q^81 + 2 * q^82 + 2 * q^85 + 2 * q^90 - 2 * q^97 + 2 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$$.

 $$n$$ $$35$$ $$37$$ $$\chi(n)$$ $$-1$$ $$i$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
47.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 −1.00000 1.00000i 0 0 1.00000i 1.00000i 1.00000 1.00000i
55.1 1.00000i 0 −1.00000 −1.00000 + 1.00000i 0 0 1.00000i 1.00000i 1.00000 + 1.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
17.c even 4 1 inner
68.f odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 68.1.f.a 2
3.b odd 2 1 612.1.l.a 2
4.b odd 2 1 CM 68.1.f.a 2
5.b even 2 1 1700.1.p.a 2
5.c odd 4 1 1700.1.n.a 2
5.c odd 4 1 1700.1.n.b 2
7.b odd 2 1 3332.1.m.b 2
7.c even 3 2 3332.1.bc.c 4
7.d odd 6 2 3332.1.bc.b 4
8.b even 2 1 1088.1.p.a 2
8.d odd 2 1 1088.1.p.a 2
12.b even 2 1 612.1.l.a 2
17.b even 2 1 1156.1.f.b 2
17.c even 4 1 inner 68.1.f.a 2
17.c even 4 1 1156.1.f.b 2
17.d even 8 2 1156.1.c.b 2
17.d even 8 2 1156.1.d.a 2
17.e odd 16 8 1156.1.g.b 8
20.d odd 2 1 1700.1.p.a 2
20.e even 4 1 1700.1.n.a 2
20.e even 4 1 1700.1.n.b 2
28.d even 2 1 3332.1.m.b 2
28.f even 6 2 3332.1.bc.b 4
28.g odd 6 2 3332.1.bc.c 4
51.f odd 4 1 612.1.l.a 2
68.d odd 2 1 1156.1.f.b 2
68.f odd 4 1 inner 68.1.f.a 2
68.f odd 4 1 1156.1.f.b 2
68.g odd 8 2 1156.1.c.b 2
68.g odd 8 2 1156.1.d.a 2
68.i even 16 8 1156.1.g.b 8
85.f odd 4 1 1700.1.n.b 2
85.i odd 4 1 1700.1.n.a 2
85.j even 4 1 1700.1.p.a 2
119.f odd 4 1 3332.1.m.b 2
119.m odd 12 2 3332.1.bc.b 4
119.n even 12 2 3332.1.bc.c 4
136.i even 4 1 1088.1.p.a 2
136.j odd 4 1 1088.1.p.a 2
204.l even 4 1 612.1.l.a 2
340.i even 4 1 1700.1.n.a 2
340.n odd 4 1 1700.1.p.a 2
340.s even 4 1 1700.1.n.b 2
476.k even 4 1 3332.1.m.b 2
476.z even 12 2 3332.1.bc.b 4
476.bb odd 12 2 3332.1.bc.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
68.1.f.a 2 1.a even 1 1 trivial
68.1.f.a 2 4.b odd 2 1 CM
68.1.f.a 2 17.c even 4 1 inner
68.1.f.a 2 68.f odd 4 1 inner
612.1.l.a 2 3.b odd 2 1
612.1.l.a 2 12.b even 2 1
612.1.l.a 2 51.f odd 4 1
612.1.l.a 2 204.l even 4 1
1088.1.p.a 2 8.b even 2 1
1088.1.p.a 2 8.d odd 2 1
1088.1.p.a 2 136.i even 4 1
1088.1.p.a 2 136.j odd 4 1
1156.1.c.b 2 17.d even 8 2
1156.1.c.b 2 68.g odd 8 2
1156.1.d.a 2 17.d even 8 2
1156.1.d.a 2 68.g odd 8 2
1156.1.f.b 2 17.b even 2 1
1156.1.f.b 2 17.c even 4 1
1156.1.f.b 2 68.d odd 2 1
1156.1.f.b 2 68.f odd 4 1
1156.1.g.b 8 17.e odd 16 8
1156.1.g.b 8 68.i even 16 8
1700.1.n.a 2 5.c odd 4 1
1700.1.n.a 2 20.e even 4 1
1700.1.n.a 2 85.i odd 4 1
1700.1.n.a 2 340.i even 4 1
1700.1.n.b 2 5.c odd 4 1
1700.1.n.b 2 20.e even 4 1
1700.1.n.b 2 85.f odd 4 1
1700.1.n.b 2 340.s even 4 1
1700.1.p.a 2 5.b even 2 1
1700.1.p.a 2 20.d odd 2 1
1700.1.p.a 2 85.j even 4 1
1700.1.p.a 2 340.n odd 4 1
3332.1.m.b 2 7.b odd 2 1
3332.1.m.b 2 28.d even 2 1
3332.1.m.b 2 119.f odd 4 1
3332.1.m.b 2 476.k even 4 1
3332.1.bc.b 4 7.d odd 6 2
3332.1.bc.b 4 28.f even 6 2
3332.1.bc.b 4 119.m odd 12 2
3332.1.bc.b 4 476.z even 12 2
3332.1.bc.c 4 7.c even 3 2
3332.1.bc.c 4 28.g odd 6 2
3332.1.bc.c 4 119.n even 12 2
3332.1.bc.c 4 476.bb odd 12 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(68, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 2T + 2$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$(T + 1)^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 2T + 2$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 2T + 2$$
$41$ $$T^{2} - 2T + 2$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$T^{2} + 2T + 2$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 2T + 2$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 2T + 2$$
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